Sommerfeld-Watson Transformation Research Articles

Reflection Characteristics of a Near-Interface Cavity in Ice at Supercritical Incidence

The long-range propagation modes in an acoustic channel under ice are basically caused by supercritical incidence. The energy distribution and transmission loss in the acoustic channel under ice are changed by a scatter in ice. The influence of a slender cylindrical cavity near and parallel to the ice-water interface on the sound propagation is analyzed using Fourier-Bessel series and Sommerfeld-Watson transformation. The research found that the acoustic field presents a beam in the mirror reflection direction at supercritical incidence, and the beam-width is proportional to secant of incident angle; meanwhile, the reflected coefficient is proportional to cosine of incident angle. The reflection coefficient increases with relative depth and Helmholtz number if the incident angle is a constant.

Computing the Scattering From Slightly Deformed Spherical Shells—A Ray-Based Approach

In a previous paper (J. A. Fawcett, IEEE J. Ocean. Eng , vol. 41, no. 3, pp. 682–688, Jul. 2016), the scattering from a spherical shell whose thickness varied with respect to the polar angle (but not azimuthally) was considered. There, it was found that the characteristics of the scattering varied significantly with respect to the monostatic angle and also with respect to the character of the thickness variation. For example, for a wave incident along the vertical axis, the amplitude of the backscattered field depended significantly upon whether a simple shell thickness profile varied from thin to thicker or vice versa. In this paper, the results from the previous paper and some new results are interpreted in terms of a ray theory. This ray theory is simply derived by considering the results from the Sommerfeld–Watson transformation for a constant thickness sphere and substituting integral averages of phase and attenuation where appropriate. For the case of the monostatic angle being off-axis (for example, broadside incidence) these integral values also vary with respect to the raypath. In this case, an integral over the set of contributing raypaths must be performed to compute the scattered pressure field. The ray-predicted scattered pressure fields are compared to those obtained from numerical benchmark computations as a function of frequency and angle of incidence.

HIGH FREQUENCY SCATTERING BY AN IMPENETRABLE SPHERE

The high frequency scattering of a scalar plane wave from an impenetrable sphere with a diameter of several thousand wavelengths is treated by the Sommerfeld-Watson transformation, the saddle-point technique (SPT), and the numerical steepest descent method (NSDM). Both the near and far flelds for the sphere are computed within the observation angle range of 0 to 180 degree. First, with the aid of the Watson transformation, the fast-convergent residue series replacing the slow-convergent Mie series is derived. Second, a new algorithm for flnding the zeros of the Hankel functions is developed. Third, a novel NSDM, which is adaptive to frequency and is hence frequency independent, is proposed to overcome the breakdown of the traditional SPT in the transition region. Numerical results show that when the observation angle is very small, the Mie series solution of the near-fleld will not be accurate due to error accumulation. Furthermore, using the proposed methods, the CPU times for both the near-fleld and far-fleld calculations are frequency independent with controllable error. This work can be used to benchmark future works for high-frequency scattering.

Computation of Wave Scattering Problems from a Spheric Body: Derivation of the New Sommerfeld-Watson Transformation

In this paper, the new Sommerfeld-Watson transformation is introduced and a new derivation processing is given. Unlike the original one, this new derivation directly starts from the solution of the traditional SommerfeldWatson transformation. The result is instantly reached using the relation between Ql(cos θ) and Pl(cos θ) which represents travelling wave and standing wave respectively. So the mathematical processing is simpler and faster than the original derivation approach. At last, the simulation results show that the new approach is in good agreement with the traditional solution.

A new Sommerfeld-Watson transformation in 3-D

A new formula on the scalar wave scattering by a sphere at high frequency is presented. A new Sommerfeld-Watson transformation is used to construct a new residue series. In the derivation, a spherical traveling wave is defined and used to reduce the complexity of the solution. Furthermore, the new solution offers a clearer physical picture.

Benchmarking an acoustic coupling theory for elastic shells of arbitrary shape

Coupling coefficients for thin shells of arbitrary curvature based upon a ray-theoretic approach devised by the authors are benchmarked against exact and high-frequency results for canonical geometries, i.e., the spherical and infinite circular cylindrical shells. The well-known high-frequency approximations of the coupling coefficients are found to describe their magnitude accurately but not the phase for midfrequencies (between ring and coincidence) when compared with results obtained by applying the Sommerfeld–Watson transformation to the exact modal series. The coupling coefficients characterize both the magnitude and the phase accurately and simply in terms of fundamental physical parameters.

Selected topics in computational ray methods

Several selected topics intimately involved with the application of ray theory to acoustic scattering are discussed. The first topic is the application of quantitative ray theory to the low-frequency acoustic scattering from large aspect ratio solids. In this approach, the scattering amplitude is developed as a time-ordered perturbation series. The series for the elastic response is explicitly summed to obtain a closed form expression that is analogous to results obtained for spherical and infinite cylindrical geometries through application of the Sommerfeld–Watson transformation. Emphasis is placed on novel features that have no counterpart on these simpler geometries (e.g., ‘‘bipolar’’ coupling phenomena) and their implications for resonance excitation. A second topic deals with a near-field/far-field target strength model for more complicated geometrical shapes that is under development at CSS. But theoretical and experimental results are presented. A third topic is the high-frequency ‘‘quasi-resonance’’ phenomenon of thin shelled structures. The structure of the ‘‘quasiresonance’’ mode is obtained for both complex wave number (real frequency) and complex frequency (real wave number) extensions of the reflection coefficient for a fluid loaded flat plate. The results are used to synthesize the scattering amplitude for spherical shells using forms previously derived for the Sommerfeld–Watson transform.

Relation between the landing-launching coefficient and the attenuation of surface waves propagating around an elastic cylindrical object

One approach for developing a physical understanding of the scattering of sound from elastic objects in water is to use the Sommerfeld–Watson transformation (SWT). Following the works of Überall etal., it is known that elastic objects resonances are due to a phase matching of the surface waves propagating around the scatterer. What is more, resonance widths are related to surface waves attenuation, thereby showing the close link between resonance and surface waves propagation phenomena. According to the geometrical theory of diffraction, surface waves are also characterized by a landing coefficient and a launching one, both identical in the present case. This landing–launching coefficient calculated from the SWT gives an indication, as resonance widths and surface waves attenuation, on the coupling between the object and the surrounding fluid. The purpose of this work is to determine an analytical relation between the landing–launching coefficient and the attenuation of surface waves. This relation comes from an identification between the Breit–Wigner approximation of resonances and the Mittag–Leffler expansions related to surface waves contribution. Thereafter, a method of identification of the background introduced by Flax etal. in the acoustic scattering theory has been developed.

Comparison of membrane wave coupling coefficients

Comparisons are made between various estimates of the coupling coefficients that characterize the longitudinal wave contribution to acoustic scattering from thin elastic circular cylindrical and spherical shells. A common method of obtaining the coupling coefficient is through the application of the Sommerfeld–Watson transformation to the exact modal series solution. From this result, a high-frequency approximation can be derived that accurately describes the magnitude, but it does not adequately represent the phase. These are compared to coupling coefficients determined using a ray-theoretic approach. This representation describes both the magnitude and the phase well when compared to that obtained from the exact modal series result using the Sommerfeld–Watson transformation. [Work supported by ONR.]

A tale of two approaches: The response of a beaded string

In one approach, the response is given by the superposition of two terms. The first is merely the response of the unbeaded infinite string. The remaining terms involve elements of the first term mingled with parameters that describe the beads; e.g., the impedance that each bead presents to the infinite string and the position of that bead on the string. In the other approach, the bays (portions of the string that lie between adjacent beads) are treated as individual dynamic systems. These dynamic systems are coupled at the beads. The model is then a cascade of coupled one-dimensional dynamic systems. In this approach, the response is a vector composed of the response in each bay. The response vector is composed of the superposition of a direct response vector and a reverberant response vector. The reverberant response itself is composed of two distinct terms. The formalisms in the two approaches appear very different although they yield identical responses when subjected to the same external drive. This identity in the responses is illustrated by computations. Unlike the Sommerfeld–Watson transformation that relates the modal and the wave approaches to describing the response of a finite string, as yet, no analogeous analytical transformation is found to relate the two approaches of concern here.

Increased accuracy in the application of the Sommerfeld–Watson transformation to scattering of acoustic waves from spherical shells

The Sommerfeld–Watson transformation (SWT) is applied to the normal mode series for scattering of an incident plane acoustic wave from a spherical elastic shell to express the scattered field in terms of its wave components. The ‘‘geometric’’ part of the scattered field is extracted and expressed as a line integral which is evaluated by shifting to a path of steepest descent (SDP). When deforming the path, one or more poles of the integrand may be crossed and their residues must be accounted for, resulting in discontinuities in the computed field unless the effects of these poles are also included in the evaluation of the SDP integral. Although the asymptotic form of the Legendre function used in the evaluation of the SDP integral is singular and invalid near the direction of backscatter, it is found that its use yields a finite and correct result for the pole correction, as well as for the saddle point approximation. This approach is applied to a thin steel shell in water using thin shell theory, and good agreement is obtained for scattering into the backward half-space between the results of the normal mode series, using shell theory, and the modified SWT calculation for ka≳3.

Relation between surface helical waves and elastic cylinder resonances

In this paper, the authors deal with theoretical and experimental studies about the acoustic scattering by an elastic solid cylinder immersed in water, at oblique incidence. Comparisons are made between the calculated form function and measured scattered pressure for incidence angles varying from 0° to 30°, in the normalized frequency range 0–30. The behavior of the resonances when the incidence angle increases is shown and the connection between helical surface waves propagation and resonance is explained, using the contributions of the resonance scattering theory and the Sommerfeld–Watson transformation. More accurately, resonances are due to the phase matching of circumnavigating helical surface waves, and a refraction effect is found to take place between the incident and the helical surface wave direction of propagation, which clearly explains the resonance shift with respect to incidence angle variations. Guided waves appear at oblique incidence with a polarization primarily transverse at small incidence angles.

Application of the Sommerfeld–Watson transformation to scattering of acoustic waves obliquely incident upon cylindrical shells

The Sommerfeld–Watson transformation (SWT) is applied to the normal mode series for scattering of an obliquely incident plane acoustic wave from an infinite circularly cylindrical shell to express the scattered field in terms of its wave components. The ‘‘geometric’’ part of the scattered field is extracted and expressed as a line integral that is evaluated by shifting to a path of steepest descent (SDP). When deforming the path, one or more poles may be crossed and their residues must be accounted for, resulting in discontinuities in the computed field unless the effects of these poles are also included in the evaluation of the SDP integral. This correction, made in the analysis in this paper, is applied to the particular example of a thin steel shell in water, using thin-shell theory. Very good agreement for scattering into the backward half-space is obtained between the results of the normal mode series and the modified SWT calculation for frequency-angle combinations such that ka cos φ≥2, where φ is the angle of incidence of the plane wave relative to the shell normal. Inclusion of the corrections to the solution also improves the calculation even when no poles are crossed by the path deformation.

Erratum: ‘‘Increased accuracy in the application of the Sommerfeld–Watson transformation to acoustic scattering from cylindrical shells’’ [J. Acoust. Soc. Am. 9 , 2739–2750 (1991)

Erratum: ‘‘Increased accuracy in the application of the Sommerfeld–Watson transformation to acoustic scattering from cylindrical shells’’ [J. Acoust. Soc. Am. <b>9</b> <b/>, 2739–2750 (1991)

Summing the spectral representations of Pöschl–Teller and Rosen–Morse fixed-energy amplitudes

The spectral representations of the fixed-energy amplitude of the symmetric and the general Pöschl–Teller potentials are summed via a Sommerfeld–Watson transformation which leads to a simple closed-form expression. The result is used to write down a similar expression for the symmetric and general Rosen–Morse potentials, exploiting the close correspondence that exists between the two systems within the Schrödinger theory and the path integral formalism via a Duru–Kleinert transformation. From the singularities of the latter amplitude the bound and continuum states of the general Rosen–Morse potential are extracted.

Increased accuracy in the application of the Sommerfeld–Watson transformation to acoustic scattering from cylindrical shells

The Sommerfeld–Watson transformation (SWT), when applied to problems of sound scattering from elastic bodies with separable geometries, allows the scattered field to be displayed in terms of waves rather than in terms of modes, and so gives a more meaningful picture of the scattering mechanisms at moderate and high frequencies. As typically applied, however, the SWT has left a discontinuity in the calculated field as a function of field point. This discontinuity is not inherent in the method, but is a result of not accounting for the effects of nearby poles in the evaluation of the line integral which represents the geometric part of the scattered field (essentially the specular term). This paper considers this effect in the particular case of broadside scattering from an infinite circular cylindrical elastic shell in an acoustic medium. The scattered field in the backward half-space, as computed from the corrected wave solution, is shown to be in excellent agreement with the field calculated from the normal mode (partial wave) series for frequencies corresponding to ka=1.5 and above, for a steel shell whose wall thickness is one percent of its radius. Shell theory, rather than the theory of elasticity, is used.

Quantitative measurement of attenuation and other parameters on biopsy samples of fresh tissue using acoustical microscopy

The Sommerfeld-Watson transformation, the reflection method and the theory of graphs are used to investigate the components of the echo-signal from a hollow elastic sphere with a concentric liquid filler, formed by reflection and rereflection at the media interfaces.

GTD for backscattering from elastic spheres and cylinders in water and the coupling of surface elastic waves with the acoustic field

The geometrical theory of diffraction (GTD) has been extended to describe surface elastic wave (SEW) contributions to backscattering from spheres and cylinders in water at high frequencies. The coupling (described by a coefficient Gl ) of the lth class of SEW with the acoustic field and the resulting contribution fl to the form function for solid spheres were previously derived [K. L. Williams and P. L. Marston, J. Acoust. Soc. Am. 79, 1702–1708 (1986)] via a Sommerfeld–Watson transformation (SWT). That work gave a Fabry–Perot representation of fl . A similar representation was postulated by applying the principles of GTD to the Lamb wave contributions to backscattering from empty cylindrical shells [V. Borovikov and N. Veksler, Wave Motion 7, 143–152 (1985)]. In either case, ‖ fl ‖=‖Gl exp[−2βl ×(π−θl )]/[1+j exp(−2πβl +i2πkac/cl )]‖ ,where j=1 for spheres and j=−1 for cylinders, each of radius a. The SEW phase velocity and attenuation coefficient are cl and βl , sin θl =c/cl , and c and k are the velocity and wavenumber in water. High-frequency resonances of the elastic response were shown to be described by the fl of the relevant SEW. The present work gives approximations for ‖Gl ‖ for spheres and cylinders in terms of βl , c/cl, and ka. These are confirmed by comparison with numerical data for ‖Gl ‖ from Borovikov et al. for cylinders and from SWT results for Rayleigh and whispering gallery waves on spheres. The ratio of the approximate ‖Gl ‖ is confirmed by geometric considerations. The analysis is applicable to other supersonic surface guided waves. The derivation uses results of resonance scattering theory and the identification of a nonresonant SEW ‘‘background.’’

Synthesis of backscattering from an elastic sphere using the Sommerfeld–Watson transformation and giving a Fabry–Perot analysis of resonances

The Sommerfeld–Watson transformation (SWT) was recently applied to the acoustic backscattering from elastic spheres in water having ka≫1 [K. L. Williams and P. L. Marston, J. Acoust. Soc. Am. 78, 1093–1102 (1985)]. Expressions for the scattering due to each class of elastic surface wave (e.g., the Rayleigh wave) were interpreted in terms of contributions from repeated circumnavigations. In the present paper, these expressions are summed in closed form as in the analysis of Fabry–Perot resonators. The form function is synthesized by adding this sum to the specular reflection. The procedure is confirmed by comparison with the exact form function f for a tungsten carbide sphere in the range 10≤ka≤80. In this case, the interference of the specular and Rayleigh contributions produces the underlying structure in ‖f(ka)‖, while the whispering gallery wave resonances produce a finer superposed structure. Phase shifts and coupling coefficients are identified which affect the signatures in ‖f(ka)‖ of the Rayleigh wave resonances.

A complex angular momentum theory of modified Coulomb scattering

The authors develop an exact complex angular momentum (CAM) theory of elastic scattering for a complex optical potential with a Coulombic tail. The present CAM theory avoids complications due to the long range nature of the Coulombic potential in a straightforward way. This is in contrast to the conventional approach in which the partial wave series is divergent and the scattering amplitude f( theta ) is usually decomposed into a pure Coulomb amplitude and a modifying convergent partial wave series. After considering some general properties of the scattering matrix element S( lambda ), the Sommerfeld-Watson transformation together with a travelling wave (near-side far-side) decomposition is used to obtain an exact representation for f( theta ) in terms of a background integral fB( theta ) and a series of subamplitudes fn(+or-)( theta ). New exact representations are derived for fB( theta ) when S( lambda ) possesses local symmetries of the type S(- lambda )=S( lambda ) exp(+or-2i pi lambda ) and S(- lambda )=S( lambda ). The subamplitudes fn(+or-)( theta ) are contour integrals and are the exact equivalents of the saddle-point integrals that arise in semiclassical theories. The fn(+or-)( theta ) can also be evaluated in terms of the Regge pole positions and residues of S( lambda ), which allows a flexible representation for f( theta ) to be derived. The exact results obtained unify the CAM theory of scattering for Coulombic and short range potentials and are especially suitable for the introduction of semiclassical approximations.